Michael Cieply for The New York Times:

The Interview generated roughly $15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday. Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all.

We gotcha covered, Mike!

Featured Comment

Stas, with a zinger for the ages:

Probably they took advice from this guy.

Angela Ensminger:

Maybe an opening question to this problem would be:. Do you think Sony had more rentals or more sales? This could lead to some interesting discussions before actually solving the problem.

[h/t Math Curmudgeon]

Blanton & Kaput modify the Christmas carol The Twelve Days of Christmas for an algebraic reasoning task befitting the season:

How many gifts did your true love receive on each day? If the song was titled “The Twenty-Five Days of Christmas,” how many gifts would your true love receive on the twenty-fifth day? How many total gifts did she or he receive on the first two days? The first three days? The first four days? How many gifts did she or he receive on all twelve days?

“The X Days of Christmas.” I like it.

Malcolm Swan, on how to begin a lesson:

Every lesson should begin by getting [students] to articulate something about what they already understand or know about something or their initial ideas. So you try and uncover where they’re starting from and make that explicit. And then when they start working on an activity, you try to confront them with things that really make them stop.

And it might be that you can do this by sitting kids together if they’ve got opposing points of views. So you get conflict between students as well as within. So you get the conflict which comes within, when you say, “I believe this, but I get that and they don’t agree.” Or you get conflict between students when they just have fundamental disagreements, when there’s a really nice mathematical argument going on. And they really do want to know and have it resolved. And the teacher’s role is to try to build a bit of tension, if you like, to try and get them to reason their way through it.

And I find the more students reason and engage like that then they can get quite emotional. But when they get through it, they remember the stuff really well. So it’s worth it.

Christopher Danielson and Megan Schmidt have both written recently and compellingly about the trouble students have when taught that math is a series of correct “steps.”

Danielson, doing his best Howard Beale:

THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.

Let me add to the conversation the category of “steps that are correct but useless.” These are great. They come from a conversation I had, like, fifteen minutes ago with a teacher named Leah Temes here at NWMC 2014.

Leah teaches Algebra II. We were talking about solving systems of equations. It’s really easy to teach the solution to a system like this as a series of correct, useful steps:

2x + 3y = 10
5x – 3y = 4

1. Add the second equation to the first one.
2. Solve for x.
3. Substitute x in either equation to solve for y.
4. Check that pair in the other equation for full credit.

Leah said she was tired of seeing her students mimic those correct steps without understanding why they worked. So instead of showing her students steps that were useful and correct, she asked them if she was allowed to add the following two equations:

2x + 3y = 10
5 = 5

To get:

2x + 3y + 5 = 10 + 5

Is everything still correct? Yes.

Was that useful? No.

This experience awakened her students to a category of steps in addition to the correct and useful ones they’re supposed to memorize and the incorrect and useless ones they’re supposed to avoid — correct and useless steps.

Alerting your students to that category of steps may make math seem less intimidating and more interesting. Math isn’t any longer a matter of staying on the right side of a line between the incorrect and correct steps. There’s another region out there, one that’s a bit less tame, a place for explorers, a place where the worst thing that can happen is you did something right but it just wasn’t useful. That category of steps also requires justification — “how do you know this is correct?” — which can help bend the student away from memorization and back towards understanding.

BTW. All of this implies a fourth category of steps — incorrect but useful. Can anybody give an example?

Featured Comments

Cathy Yenca:

I do a similar thing when solving equations in one variable by asking students if I can add 1,000,000, let’s say, to each side of an equation… or if I can subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.

David Petro:

I have long been a fan of deliberately sabotaging a solution to something that I might be doing on the board so that somewhere down the road things become obviously wrong. This is so students can start to develop strategies for what to do when this happens.

Many will tell you that it’s important for students to make mistakes (in fact, that they learn the most when they do). But that sometimes runs counter to what they see in class. That is, a teacher demonstrating flawless execution of mathematics. Even some of our best students often won’t even attempt a problem unless they are sure they will get it correct. If they are ever going to become comfortable with making mistakes as part of the normal process then we have to include managing those mistakes as part of our day to day in class.

Moana Evans:

[It’s] incorrect but useful to estimate things like area problems, in order to find out a ballpark figure and check if you’ve done the math right.